Anticipating synchronization with machine learning

Abstract

In applications of dynamical systems, situations can arise where it is desired to predict the onset of synchronization as it can lead to characteristic and significant changes in the system performance and behaviors, for better or worse. In experimental and real settings, the system equations are often unknown, raising the need to develop a prediction framework that is model free and fully data driven. We contemplate that this challenging problem can be addressed with machine learning. In particular, exploiting reservoir computing or echo state networks, we devise a "parameter-aware" scheme to train the neural machine using asynchronous time series, i.e., in the parameter regime prior to the onset of synchronization. A properly trained machine will possess the power to predict the synchronization transition in that, with a given amount of parameter drift, whether the system would remain asynchronous or exhibit synchronous dynamics can be accurately anticipated. We demonstrate the machine-learning based framework using representative chaotic models and small network systems that exhibit continuous (second-order) or abrupt (first-order) transitions. A remarkable feature is that, for a network system exhibiting an explosive (first-order) transition and a hysteresis loop in synchronization, the machine learning scheme is capable of accurately predicting these features, including the precise locations of the transition points associated with the forward and backward transition paths.

Learning Hamiltonian dynamics by reservoir computer

“…the vector βb) in Eq. ( 1) [19,20,33]. The adoption of the parameter-aware RC is motivated by the fact that in many realistic situations only the time series of a limited number of system motions are available, while the mission is to predict the dynamics, or replicate the system climate, of many unknown motions.…”

“…We adopt the architecture of parameter-aware RC to learn the dynamics of Hamiltonian systems [19,20]. In this architecture, the RC is constituted by four modules: the I/R layer (input-to-reservoir), the parameter-control module, the reservoir network, and the R/O layer (reservoir-to-output).…”

“…Besides predicting chaos evolutions, RC has also been exploited to address other long-standing questions in nonlinear science, saying, for example, reconstructing chaotic attractors and calculating Lyapunov exponents [17], synchronizing chaotic oscillators [8,18], predicting system collapses [19], reconstructing synchronization transition paths [20], transferring knowledge between different systems [21,22], to name just a few. These studies, while demonstrating the power of RC in solving different nonlinear questions, also give insights on the working mechanisms of RC.…”

“…This ability, knowing as climate replication, suggests that it is the intrinsic dynamics of the chaotic system * Email address: wangxg@snnu.edu.cn that RC essentially learns from the data, instead of the mathematical expressions describing the time series. Exploiting this ability, model-free techniques have been proposed in recent years to predict the bifurcations in nonlinear dynamical systems, e.g., reproducing the bifurcation diagram of classical chaotic systems [13,14], anticipating the critical points of system collapse [19], predicting the critical coupling for synchronization [20], etc. Another property revealed recently in exploiting RC is that knowledge can be transferred between different dynamical systems, namely the ability of transfer learning [21,22].…”

“…Our main objective in the present work is just to verify this speculation. To be specific, by a recently proposed architecture of parameter-aware RC [19,20], we train RC by the time series of Hamiltonian systems acquired at a handful of parameters, and then use the trained RC to replicate the dynamics climates. We are able to show that the trained RC not only is able to replicate the climates associated with the training parameters, but also is able to generate the entire KAM dynamics diagram with a high precision by tuning a control parameter externally.…”